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The Double-Angle Formulas - Problem 1
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
So let’s recall the double angle identities; sine of 2 theta equals 2 sine theta cosine theta and cosine 2 theta equals cosine squared theta minus sine squared theta. Let’s use this on our problem.
Problem says use sine of 60 degrees and cosine of 60 degrees to find the sine of 120 and the cosine of 120. Here we could use other methods to find the sine and cosine of 120 but let’s use the double angle formulas.
First sine the sine of 120 degrees is the sine of double 60 degrees. Sine of 2 times an angle is twice the sine of the angle times the cosine of the angle and that’s 2 times sine of 60 which is root 3 over 2 cosine 60 which is a half. Now our 2 is going to cancel and we are left with root 3 over 2. Now we're at the cosine of 120 in the middle room we’ll come back to this, cosine of 120.
Cosine squared minus sine squared, cosine squared of 60 degrees minus sine squared of 60 degrees. Now remember when you see the superscript too it’s the whole cosine that’s getting squared. The cosine of 60 is a half so that whole thing gets squared and the sine of 60 root 3 over 2 and that whole thing gets squared. And so we’ve got 1/2 squared is a 1/4 minus root 3 over 2 squared is 3/4. 1/4 minus 3/4, -1/2 so the cosine of 120 is -1/2.
What about part b sine of 240 and cosine of 240 you can actually keep doing this indefinitely. Sine 240 is sine of 2 times 120 so we can take our answers from before this is 2 sine 120 cosine 120 and that’s 2 times the sine of 120 was root 3 over 2 and the cosine of 120 was -1/2. 2’s cancel and we get negative root 3 over 2. What about the cosine of 240? Cosine 240 cosine of 2 times 120 so we need our cosine double angle formula again it's cosine squared minus sine squared. Cosine squared of 120 minus sine squared of 120, now the cosine of 120 was -1/2. So that’s -1 squared minus sine of 120 root 3 over 2, and that will be squared. -1/2 squared is a quarter minus root 3 over 2 squared is 3/4 so 1/4 minus 3/4 is -1/2, that’s it.
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